L(s) = 1 | + (0.965 + 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (−1.99 + 1.01i)5-s + (0.499 − 0.866i)6-s + (−0.219 − 0.219i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.18 + 0.465i)10-s + 6.33·11-s + (0.707 − 0.707i)12-s + (3.59 − 0.962i)13-s + (−0.155 − 0.268i)14-s + (0.465 + 2.18i)15-s + (0.500 + 0.866i)16-s + (0.885 − 3.30i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (−0.890 + 0.454i)5-s + (0.204 − 0.353i)6-s + (−0.0828 − 0.0828i)7-s + (0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.691 + 0.147i)10-s + 1.90·11-s + (0.204 − 0.204i)12-s + (0.996 − 0.267i)13-s + (−0.0414 − 0.0717i)14-s + (0.120 + 0.564i)15-s + (0.125 + 0.216i)16-s + (0.214 − 0.801i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21262 - 0.0237916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21262 - 0.0237916i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (1.99 - 1.01i)T \) |
| 19 | \( 1 + (-3.97 - 1.79i)T \) |
good | 7 | \( 1 + (0.219 + 0.219i)T + 7iT^{2} \) |
| 11 | \( 1 - 6.33T + 11T^{2} \) |
| 13 | \( 1 + (-3.59 + 0.962i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.885 + 3.30i)T + (-14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (-0.601 - 2.24i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.42 - 5.93i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.71iT - 31T^{2} \) |
| 37 | \( 1 + (-0.817 + 0.817i)T - 37iT^{2} \) |
| 41 | \( 1 + (8.81 - 5.08i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.44 + 2.26i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (10.1 - 2.73i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.96 + 1.06i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.367 - 0.636i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.17 + 3.76i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.33 + 4.98i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.14 - 0.663i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.65 - 1.24i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.21 - 12.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.190 - 0.190i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.18 + 8.97i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.9 + 4.26i)T + (84.0 + 48.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.31510486565505405617873878328, −9.846983250117558172657455053154, −8.770700307079163507536572363513, −7.87803958567129329610836419364, −6.92752442073791454949781812775, −6.45500086714140129043792864879, −5.17095262211313942714408155259, −3.72590157040067348262454303205, −3.32961324162913203876089498862, −1.39540547128346122896817246017,
1.41162700999618990440542151772, 3.43842845463303012714149577929, 3.91612016688123667737156674441, 4.87225841808326873245531365409, 6.09776482671504660401380702241, 6.96013180956637083567624094106, 8.270058412482333830381923052055, 8.969832350996579293132784732308, 9.862726958058077351010228148813, 11.05932787723645112520826529536